91Ó°ÊÓ

The quadratic formula is a vital tool in algebra for solving quadratic equations. Quadratic equations are in the standard form \( ax^2 + bx + c = 0 \), where \( a \), \( b \), and \( c \) are constants. This formula gives us a way to find the values of \( x \) that satisfy the equation. The formula is: \[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \] It enables us to solve any quadratic equation by substituting the values of \( a \), \( b \), and \( c \) directly into it.

• The symbol \( \pm \) indicates that there are usually two solutions, corresponding to the addition and subtraction of the square root term.
• The formula relies on the discriminant \( b^2 - 4ac \) which determines the nature of the solutions.

In our example, \( a = 1 \), \( b = -1 \), and \( c = -\frac{11}{4} \) enabled us to find solutions using these values. Using the quadratic formula helps us solve various real-world problems where quadratic equations appear.

The discriminant is a crucial part of the quadratic formula. It is represented as \( D = b^2 - 4ac \), where \( a \), \( b \), and \( c \) are coefficients from the quadratic equation \( ax^2 + bx + c = 0 \). The value of the discriminant tells us about the nature of the roots of the equation:

• If \( D > 0 \), there are two distinct real roots.
• If \( D = 0 \), there is exactly one real root (also known as a repeated root).
• If \( D < 0 \), there are no real roots; instead, the roots are complex numbers.

In our problem, the discriminant \( D = 12 \) was calculated using \( b = -1 \) and \( a = 1, c = -\frac{11}{4} \). Since \( D = 12 \) which is greater than zero, it confirmed the presence of two distinct real roots for the equation.

Real roots in the context of a quadratic equation refer to solutions that are real numbers, as opposed to complex numbers. When solving quadratic equations, determining the type of roots is important:

• When the discriminant \( D > 0 \), as in our example, there are two distinct real roots. This means the parabola represented by the quadratic equation touches the x-axis at two points.
• If \( D = 0 \), there is exactly one real root, meaning the parabola is tangent to the x-axis.
• Conversely, if \( D < 0 \), there are no real roots, as the parabola does not intersect the x-axis.

For the equation \( x^2 - x - \frac{11}{4} = 0 \), solving yielded two distinct real roots, approximately \( x = 2.23 \) and \( x = -1.23 \). These values show where the equation reaches zero or where the curve crosses the x-axis, highlighting important intersections.